huneynumb

Sally constructed a dart board. The possible scores are 0,1,2,3. A score of 0 is obtained if the dart misses the board. Sally challenes John to a game consisting of each player throwing 6 darts at the board. The scores from the 6 throws are added. In how many ways can a total score of 15 or 16 be obtained?

Note that the thw total score obtained from 1+0+0+3+3+2 is consisdered different to the score 0+0+1+3+3=2

a) How many times must a die be thrown to be sure that the same number occurs twice?

b) How many times must two die be thrown to be sure that the same total score occurs at least six times?

c) How many times must n dice be thrown to be sure that the same total score occurs at least p times?

A car, a van, a truck and a bike are all travelling in the same direction on the same road, each at its own constant speed. At 10 am, the car overtakes the van; at noon, its overtakes the truck; at

2pm its overtakes the bike. At 4 pm the truck overtakes the bike and, at 6pm, the van overtakes the truck.

a) Let c an T represent the speeds in km/h of the car and the truck

respectively.

i) Find the speeds of the van and the bike in terms of c and T

ii) Show that the time when the van overtakes the bike is the same

regardless of the speeds of the car and the truck

When 391758 and 394915 are divided by a certain three digit number, the three digit remainder is the same in each case. Find the divisor.

Find all two digit-numbers N such that N+2 is also a two-digit number and the digit sum of N+2 is less than the digit sum of N

Express 1994 as a sum of consecutive positive integers and show that this is the only way to do it.

(a) Find the largest four digit number having exactly three factors, including 1 and itself.

(b) Find the largest three digit number which has exactly ten factors including 1 and itself.

Express 1991 as a sum of consecutive positive integers and show that this is the only way to do it.

What are the last two digits of

a) 3^1994

b) 7^1994

c) 3^1994 + 7^1994

d) 7^1994 - 3^1994

But what do you mean by "A number p^v*q^b*r^n*... (standard prime factorization with uncommon notation) have (v + 1)(b + 1)(n + 1) divisors. We are seeking 6 = 2*3 (a*b*..., a,b,... > 1) divisors" and how did you get the answers

(a) Find all positive integers N such that the product 2005 x N has exactly six divisors

(b) Find all composite integers M such that the product 2005 x M has exactly eight divisors